Two remarks on frequent hypercyclicity

We show that if T : X → X is a continuous linear operator on an F -space X ≠ { 0 } , then the set of frequently hypercyclic vectors of T is of first category in X , and this answers a question of A. Bonilla and K.-G. Grosse-Erdmann. We also show that if T : X → X is a bounded linear operator on a Banach space X ≠ { 0 } and if T is frequently hypercyclic (or, more generally, syndetically transitive), then the T ∗ -orbit of every non-zero element of X ∗ is bounded away from 0, and in particular T ∗ is not hypercyclic.